June 22, 2026 - Calculus 2- Exam Review Applications of Integration and Differential Equations

These vocabulary flashcards cover differential equations, physics applications like work and density, and late-transcendental calculus concepts including arc length and volumes of revolution based on the lecture transcript.

Separable Equation

A differential equation where the variables can be rearranged so that all terms involving $y$ are on one side and all terms involving $x$ are on the other, allowing it to be solved by integration.

Initial Condition

A specific value for the dependent variable at a designated starting time, such as $y(0) = 0$, used to determine the constant $C$ in a differential equation's solution.

Limiting Amount

The value that the amount of a substance, like sugar, approaches as time $t$ goes to infinity, calculated using a limit.

Inflow Minus Outflow

The formula used to set up differential equations for mixing problems, represented as $\frac{ds}{dt} = \text{rate in} - \text{rate out}$.

Concentration

The ratio of a substance's mass to the total volume of the liquid in a tank, often expressed in units like $kg/L$.

Work ($W$)

The measure of energy transfer that occurs when an object is moved by a force, defined mathematically as the integral of force times displacement: $W = \text{force} \times \text{distance}$.

Weight Density of Water

A constant used in work problems involving liquids, specified in the lecture as approximately $62.4\text{ lb/ft}^3$.

Arc Length

The distance along a curve, calculated using the integral formula $\text{Arc Length} = \text{integral of } \text{square root of } 1 + (\frac{dy}{dx})^2 dx$.

Surface Area of Solid Revolution

The area of the surface generated by rotating a curve about an axis, calculated by multiplying the arc length by the circumference of the rotation circle ($2\text{ }\text{pi}\text{ }\times\text{ radius}$).

Radial Mass Density

The distribution of mass in a circular object where density changes from the center radius out to a fixed edge radius $R$.

Equilibrium Solution

A value of $y$ for which the derivative $\frac{dy}{dx}$ equals zero, such as $y = 4$ or $y = -2$ in specific differential equations.

Disk Method Volume

The volume of a solid of revolution found by integrating the cross-sectional area of circular disks, defined as $\text{Volume} = \text{integral of } \text{pi} \times [f(x)]^2 dx$.

Washer Method Volume

The volume of a solid of revolution with a hole in the center, calculated as $\text{Volume} = \text{integral of } \text{pi} \times (\text{outer radius}^2 - \text{inner radius}^2) dx$.

Point-Slope Form

A linear equation form used to find the radius of an inverted cone during work problems, given by $y - y_1 = m(x - x_1)$, where $m$ is the slope.

Circumference of a Circle

The distance around the edge of a circle, used in radial density and surface area problems, expressed as $2\text{ pi } \times x$.

Exponential Form

The state of an equation after converting from a natural logarithm, such as transforming $\text{ln}(|y|) = f(x)$ into $|y| = e^{f(x)}$.

Weight Density Constant for Cubic Feet

The value $62.4$ representing the pounds per cubic feet needed to lift water against gravity.